Homogenization of a system of second-order differential equations is performed in the case of a nonuniformly perforated rectangle where the sizes of the holes and the distances between pairs of them decrease as the distance from one of the bases of the rectangle increases. The Neumann conditions are assumed on the boundaries of the holes. The formal asymptotics of the solution is constructed, which involves the usual Ansatz of homogenization theory and also some Ansatze typical of solutions of boundary-value problems in thin domains, in particular, exponential boundary layers. Justification of the asymptotics is done with the help of the Korn inequality, which is proved for the perforated domain Ω(h). Depending on the properties of the right-hand side, the norm of the difference between the true and the approximate solutions in the Sobolev space H1(Ω(h)) is estimated by the quantity chκ with κ ∈ (0, 1/2]. §1. Setting of the problem and description of results 1. Geometry of the domain. We start with introducing a fractal-type perforated domain for which the sizes of the holes and the distances between pairs of holes decrease when moving away from the base. For this, we consider periodicity cells of different sizes. Let Ξ = (0, b1) × (0, b2) be a rectangle. Also, let ω ∈ R be a domain having smooth boundary ∂ω and such that the closure ω = ω∪∂ω lies in the interior of Ξ. We denote by Ξ the result of contraction of Ξ along the x1-axis with the coefficient q−j and split the rectangle Ξ, which turns out to be thin for large j, into q equal rectangles Ξj of size b1q −j × b2q ; it is assumed that q, j ∈ N := {1, 2, . . . } and the number q > 1 is fixed. Here and in the sequel, a lower index corresponds to contraction in two directions, and an upper index symbolizes one-directional contraction (along the x1-axis). Replacing each of the small rectangles Ξj with the cell Sj = Ξj ωj , we obtain a cell S that consists of q equal cells Sj and is a thin rectangle with small holes ωj (cf. Figure 1). Now we describe the perforated domain itself. We consider another rectangle Q = [0, a1]× [0, a2] and introduce a small parameter h > 0 so that ai = Nibih, where the Ni are large integers, i = 1, 2. The segments Γj = {x = (x1, x2) : x1 ∈ (0, a1), x2 = γj}, where γj = jhb2, split Q into strips Π1(h), . . . ,ΠN2(h) of width hb2. Each of these strips is divided into congruent closed rectangles Q p (h) Q(h) of size hqb1 × hb2; here q ∈ N and p = 1, . . . , qN1. If q = 1, then all rectangles are equal, but for q ≥ 2 the bases of the rectangles become shorter as j grows, and the partition acquires an anisotropic fractal structure (Figure 1). 2000 Mathematics Subject Classification. Primary 35J99.